In a small bowl, sprinkle the yeast over the lukewarm water. Stir briefly to combine and allow to sit at room temperature until frothy, about 10 minutes.
In the bowl of a stand mixer fitted with a dough hook or a large bowl, combine the flour, granulated sugar, and salt.
Place the cooked sweet potato in the bowl of a food processor and add the lukewarm milk. Pulse, scraping down the sides of the bowl as needed, to form a creamy, smooth mixture.
Slowly pour the frothy yeast water mixture into the flour, followed by the sweet potato puree, softened butter, and egg to form a soft, smooth, and slightly sticky dough.
On a lightly floured surface, knead the dough until smooth and elastic, then transfer to a large bowl, cover with plastic wrap or a towel, and allow to rest at room temperature until doubled, 1-2 hours.
Grease a 9 x 13 inch (23 x 33 centimeter) baking dish with butter.
On a lightly floured surface, divide the dough into 12 equal pieces.
Form each piece into a smooth ball, seam side down, and arrange in the greased baking dish about 1 inch (2.5 centimeters) apart.
Cover the baking dish with a towel and allow to rise at room temperature until puffed, about 30 minutes.
Preheat oven to 375˚F (190˚C) and place the oven rack in the center.
In a small bowl, beat together the egg yolk and milk.
Gently brush the egg wash over the risen rolls.
Transfer the baking dish to the preheated oven and bake until the rolls are golden, 12-15 minutes.
Remove the baking dish from the oven and brush with the melted butter.
These Sweet Potato Rolls are best served warm.